Материал: Демирчян, Нейман, Коровкин, Чечурин. Теоретические основы электротехники.т.2.2003
2.
à
) Ïðèìåíÿÿ ìåòîä íàëîæåíèÿ, ïðåäñòàâëÿåì íàïðÿæåíèå â âèäå ñóììû äâóõ
ñêà÷êîîáðàçíûõ íàïðÿæåíèé
U
0
(
t – t
1
) è –
U
0
(
t – t
2
) è çàïèñûâàåì èñêîìîå èçî-
áðàæåíèå
U
(
p
)
=
U
p
e
U
p
e
U
p
e
e
pt
pt
pt
pt
0
0
0
1
2
1
2
-
-
-
-
-
=
-
(
), ó÷èòûâàÿ, ÷òî èçîáðàæå-
íèå ñêà÷êîîáðàçíîãî íàïðÿæåíèÿ
U
0
åñòü
U
0
/
p
.
Âû÷èñëÿÿ èíòåãðàë Ëàïëàñà, ïîëó÷àåì òîò æå ðåçóëüòàò:
U p
u t e dt U
e dt U
p
e
e
pt
pt
t
t
pt
pt
( )
( )
(
).
=
=
=
-
-
¥
-
-
-
ò
ò
0
0
0
1
2
1
2
Åñëè äëèòåëüíîñòü
t
2
–
t
1
= D
t
äåéñòâèÿ èìïóëüñà íàïðÿæåíèÿ ñòðåìèòñÿ ê íóëþ,
à åãî àìïëèòóäà âîçðàñòàåò, òàê ÷òî ïðîèçâåäåíèå
U
0
×D
t
ñîõðàíÿåòñÿ ïîñòîÿííûì,
òî â ïðåäåëå ïðè
D
t
®
0 èçîáðàæåíèå ïðèíèìàåò âèä
U p
U
p
e
e
t
t U te
t
t
pt
pt
pt
( )
lim
.
=
-
=
®
-
-
-
0
0
2
1
1
2
1
D
D
D
3, 4.
Âîñïîëüçóåìñÿ èçîáðàæåíèåì ïðîèçâîäíîé
f
¢
(
t
) ôóíêöèè
f
(
t
):
f t e dt
pt
'
( )
-
¥
ò
0
=
pF
(
p
) –
f
(0).
Ïðè
p
® ¥
èìååì lim
p
®¥
pF
(
p
)
=
lim
p
®¥
[
f
(0) +
f t e dt
pt
'
( )
-
¥
ò
0
]
=
f
(0), à ïðè
p
®
0
lim
( )
( )
( )
( )
p
pF p
f
f t
f
®
¥
=
+
=
¥
0
0
0
.
5.
à
)
i
(+0)
=
0,
i
(
¥
)
=
U
0
/2
r
;
á
)
i
(+0)
=
U
0
/
r
,
i
(
¥
)
=
0;
â
)
i
(+0)
=
U
0
/
r
,
i
(
¥
)
=
U
0
/
r
;
ã
)
i
(+0)
=
0,
i
(
¥
)
=
0;
ä
)
i
(+0)
=
U
0
/
r
,
i
(
¥
)
=
0.
10.2. Ðàñ÷åò ïåðåõîäíûõ ïðîöåññîâ îïåðàòîðíûì ìåòîäîì
ÂÎÏÐÎÑÛ
1.
Ïðè ðàñ÷åòå ïåðåõîäíûõ ïðîöåññîâ îïåðàòîðíûì ìåòîäîì òîêè â êàòóøêàõ
èíäóêòèâíîñòè è íàïðÿæåíèÿ íà êîíäåíñàòîðàõ â ìîìåíò âðåìåíè
t
=
+0 âõîäÿò
â óðàâíåíèÿ âòîðîãî çàêîíà Êèðõãîôà è ó÷èòûâàþòñÿ, òàêèì îáðàçîì, íà íà÷àëü-
íîé ñòàäèè ðåøåíèÿ çàäà÷è, à èìåííî íà ýòàïå ñîñòàâëåíèÿ óðàâíåíèé â îïåðà-
òîðíîé ôîðìå. Ïðè ðàñ÷åòå ïåðåõîäíûõ ïðîöåññîâ êëàññè÷åñêèì ìåòîäîì íà-
÷àëüíûå óñëîâèÿ (òîêè â êàòóøêàõ è íàïðÿæåíèÿ íà êîíäåíñàòîðàõ) ó÷èòûâàþò
óæå ïîñëå ñîñòàâëåíèÿ è ðåøåíèÿ óðàâíåíèé, à èìåííî íà ýòàïå íàõîæäåíèÿ
ïîñòîÿííûõ èíòåãðèðîâàíèÿ, âõîäÿùèõ â ðåøåíèå äèôôåðåíöèàëüíûõ óðàâíå-
íèé öåïè.
2.
 îáùåì ñëó÷àå ïðè íåíóëåâûõ íà÷àëüíûõ óñëîâèÿõ òîê
I
(
p
) íà âõîäå äâóõïî-
ëþñíèêà íåëüçÿ ïðåäñòàâèòü â âèäå
I
(
p
)
=
U
(
p
)/
Z
(
p
), ãäå
U
(
p
) — îïåðàòîðíîå
èçîáðàæåíèå âõîäíîãî íàïðÿæåíèÿ. Ïîýòîìó îïåðàòîðíîå ñîïðîòèâëåíèå
Z
(
p
)
íåëüçÿ ïîëó÷èòü, çàìåíèâ â âûðàæåíèè
Z
(
j
w
) êîìïëåêñíîãî ñîïðîòèâëåíèÿ
äâóõïîëþñíèêà âåëè÷èíó
j
w
íà îïåðàòîð
p
. Îäíàêî ïðè íóëåâûõ íà÷àëüíûõ
Îòâåòû íà âîïðîñû, ðåøåíèÿ óïðàæíåíèé è çàäà÷
503
óñëîâèÿõ èç âûðàæåíèÿ
I
(
p
)
=
U
(
p
)/
Z
(
p
) ìîæíî íàéòè îïåðàòîðíîå ñîïðîòèâëå-
íèå
Z
(
p
)
=
U
(
p
)/
I
(
p
), êîòîðîå ñîâïàäàåò ñ êîìïëåêñíûì ñîïðîòèâëåíèåì
Z
(
j
w
)
ïîñëå çàìåíû â íåì âåëè÷èíû
j
w
íà îïåðàòîð
p
.
4.
Ïðè íóëåâûõ íà÷àëüíûõ óñëîâèÿõ óðàâíåíèÿ çàêîíîâ Êèðõãîôà ìîæíî ïîëó-
÷èòü, çàìåíÿÿ â ñîîòâåòñòâóþùèõ óðàâíåíèÿõ êîìïëåêñíîãî ìåòîäà êîìïëåêñ-
íûå òîêè
&
I
, íàïðÿæåíèÿ
&
U
, ñîïðîòèâëåíèÿ
Z
è ïðîâîäèìîñòè
Y
íà âåëè÷èíû
I
(
p
),
U
(
p
),
Z
(
p
),
Y
(
p
). Îáðàòíîå òàêæå ñïðàâåäëèâî: çàïèñàâ óðàâíåíèÿ çàêîíîâ Êèðõ-
ãîôà, ìîæåì, ïðèíèìàÿ íà÷àëüíûå óñëîâèÿ íóëåâûìè, ïåðåéòè ê óðàâíåíèÿì â
êîìïëåêñíîé ôîðìå ïîñëå çàìåíû â íèõ îïåðàòîðà
p
íà âåëè÷èíó
j
w
.
7.
Åñëè ïîëèíîì
H
(
p
) îïåðàòîðíîãî èçîáðàæåíèÿ òîêà
I
(
p
)
=
G
(
p
)/
H
(
p
) íå èìååò
êîìïëåêñíûõ êîðíåé, òî âñå êîýôôèöèåíòû
G
(
p
)/
H
(
p
) âåùåñòâåííûå. Åñëè æå
îí èìååò õîòÿ áû îäíó ïàðó êîìïëåêñíî-ñîïðÿæåííûõ êîðíåé
p
1
=
–
a
+
j
w
è
p
2
=
–
a
–
j
w
, òî êîýôôèöèåíòû
À
(
ð
1
)
=
G
(
p
1
)/
H
¢
(
p
1
),
A
(
p
2
)
=
G
(
p
2
)/
H
¢
(
p
2
), âõîäÿ-
ùèå â âûðàæåíèå èñêîìîãî òîêà
i
(
t
)
=
exp (–
a
t
)[
A
(
p
1
) exp (
j
w
t
) +
A
(
p
2
) exp(–
j
w
t
)]
â îáùåì ñëó÷àå, êîãäà
A
(
p
1
)
¹
A
(
p
2
), äîëæíû áûòü êîìïëåêñíûìè, òàê êàê òîëüêî
òîãäà òîê
i
(
t
) áóäåò âåùåñòâåííûì.
8.
Èç óðàâíåíèÿ
Z
(
p
)
I
(
p
)
=
U
(
p
), â êîòîðîì
I
(
p
) — òîê íà âõîäå äâóõïîëþñíèêà,
Z
(
p
) — åãî îïåðàòîðíîå ñîïðîòèâëåíèå, ñëåäóåò, ÷òî ïîëèíîì
Z
(
p
) ñîâïàäàåò ñ
õàðàêòåðèñòè÷åñêèì ïîëèíîìîì ñîîòâåòñòâóþùåãî äèôôåðåíöèàëüíîãî óðàâíå-
íèÿ öåïè. Äåéñòâèòåëüíî, ïåðåõîä îò äèôôåðåíöèàëüíîãî óðàâíåíèÿ öåïè ê õà-
ðàêòåðèñòè÷åñêîìó âûïîëíÿåòñÿ òàê æå, êàê è ïåðåõîä îò äèôôåðåíöèàëüíîãî
óðàâíåíèÿ ê îïåðàòîðíîìó, à èìåííî ïóòåì çàìåíû
k
-é ïðîèçâîäíîé òîêà íà
k
-þ
ñòåïåíü âåëè÷èíû
a
(ïðè ñîñòàâëåíèè õàðàêòåðèñòè÷åñêîãî óðàâíåíèÿ) ëèáî íà
k
-þ ñòåïåíü îïåðàòîðà
p
ïðè ïîëó÷åíèè îïåðàòîðíîãî óðàâíåíèÿ.  ïîñëåäíåì
ñëó÷àå íà÷àëüíûå óñëîâèÿ ìîæíî âñåãäà ïðèíÿòü íóëåâûìè, òàê êàê ñâîéñòâà
öåïè íå çàâèñÿò îò çàäàâàåìûõ íà÷àëüíûõ óñëîâèé. Ïðè ðàñ÷åòå îïåðàòîðíîãî
ñîïðîòèâëåíèÿ öåïè òîêè âñåõ êàòóøåê èíäóêòèâíîñòè è íàïðÿæåíèÿ íà âñåõ
êîíäåíñàòîðàõ ñëåäóåò ïðèíÿòü ðàâíûìè íóëþ, âñå èñòî÷íèêè ÝÄÑ äîëæíû áûòü
çàìêíóòû íàêîðîòêî, à âåòâè ñ èñòî÷íèêàìè òîêà — ðàçîðâàíû.
ÓÏÐÀÆÍÅÍÈß
1.
Îïåðàòîðíûå ñîïðîòèâëåíèÿ öåïåé ñîîòâåòñòâóþùèõ âàðèàíòîâ ðàâíû:
à
)
Z p
r r
pL r
r
r
pL
( )
(
)
=
+
+
+
1 2
1
2
2
;
á
)
Z p
r pL p rLC
rCp
( )
= +
+
+
2
1
;
â
)
Z p
r pL p rLC
r pL Cp
( )
(
)
= +
+
+
2
;
ã
)
Z p
pL prC
Cp r pL
( )
(
)
(
)
=
+
+
+
1
1
;
ä
)
Z p
r pL
r pL pC
( )
(
)
=
+
+
+
1
;
å
)
Z p
r pL p rLC
p LC
( )
= +
+
+
2
2
1
.
2.
Èçîáðàæåííûì íà ðèñ. Ð10.1 ñõåìàì ýëåêòðè÷åñêèõ öåïåé ñîîòâåòñòâóþò çà-
ïèñàííûå â îïåðàòîðíîé ôîðìå óðàâíåíèÿ çàêîíîâ Êèðõãîôà:
à
)
I
r
(
p
)
=
I
L
(
p
) +
I
C
(
p
),
rI
r
(
p
) +
pLI
L
(
p
)
=
E
(
p
) +
Li
L
(0),
pLI
L
(
p
) – 1
Cp
I
C
(
p
)
=
=
Li
L
(0) +
E
p
+
u
p
C
( )
0 .
504
Îòâåòû íà âîïðîñû, ðåøåíèÿ óïðàæíåíèé è çàäà÷
á
)
I
C
(
p
)
= Á
(
p
) +
I
L
(
p
),
pLI
L
(
p
) + 1
Cp
I
C
(
p
)
=
Li
L
(0) –
u
p
C
( )
0 .
ä
)
I
L
(
p
)
=
I
C
(
p
) +
I
r
1
(
p
),
Á
(
p
) +
I p
I p
r
r
1
2
( )
( )
=
,
I
L
(
p
)
pL
+
I
C
(
p
) 1
Cp
=
=
E
p
+
Li
L
(0) –
u
p
C
( )
0 ,
I
r
1
(
p
)
r
1
+
I
r
2
(
p
)
r
2
–
I
C
(
p
) 1
Cp
=
u
p
C
( )
0 .
4.
Ó÷èòûâàÿ, ÷òî îïåðàòîðíîå èçîáðàæåíèå ÝÄÑ è òîêà çàäàííûõ èñòî÷íèêîâ
cóòü
E p
E
p
m
( )
=
+
w
w
2
2
,
Á
= Á
+
( )
p
p
m
w
w
2
2
, ïîëó÷àåì:
á
)
I p
E
r
r Cp
p
r r Cp
m
( )
[(
)
]
(
) (
)
=
+
+
+
+
w
w
1
2
2
2
1
2
1
1
;
â
)
I p
E Cp
p
p LC
m
( )
(
)(
)
=
+
+
w
w
2
2
2
1
;
ã
)
I p
E
r
r
pL
p
r pL r
m
( )
(
)
(
) (
)
=
+ +
+
+
w
w
1
2
2
2
1
2
;
ä
)
U p
J r
p
prC
m
( )
(
)(
)
=
+
+
w
w
2
2
1
;
å
)
U p
rpL
p
pL r
m
( )
(
)(
)
=
Á
+
+
w
w
2
2
;
ç
)
I p
E
pL r
p
r
r pL r r
m
( )
(
)
(
)[(
)
]
=
+
+
+
+
w
w
2
2
1
2
1 2
.
5.
Ó÷èòûâàÿ çàäàííûå íà÷àëüíûå óñëîâèÿ
i
L
(–0)
=
0,
u
C
(–0)
=
U
0
=
120 Â ïðè ñî-
ñòàâëåíèè óðàâíåíèé çàêîíîâ Êèðõãîôà â îïåðàòîðíîé ôîðìå
-
+
+
=
I p
I p
I p
1
2
3
0
( )
( )
( )
,
rI p
I p
Cp
U
p
u
p
C
1
2
0
0
( )
( )
( )
+
=
-
,
I p pL I p
Cp
u
p
C
3
2
0
( )
( )
( )
-
=
è ðåøàÿ èõ îòíîñèòåëüíî òîêà
I
2
(
p
), ïîëó÷àåì:
I p
u
rC
p rLC pL r
p
p
C
2
2
2
5
2
0
48 10
4 10
0 1
40
( )
( )
,
,
= -
+
+
=
-
×
×
+
+
-
-
.
Ðåøàÿ óðàâíåíèå
H
(
p
)
=
0, ïîëó÷àåì
p
1
=
–2000,
p
2
=
–500. Äàëåå íàõîäèì
G
(
p
1
)
=
=
G
(
p
2
)
=
–4,8
×
10
–2
,
H
¢
(
p
1
)
=
–0,06,
H
¢
(
p
2
)
=
0,06 è òîê
i
2
(
t
)
=
0,8
e
–2000
t
– 0,8
e
–500
t
À.
Äàëåå ïîëó÷àåì:
Îòâåòû íà âîïðîñû, ðåøåíèÿ óïðàæíåíèé è çàäà÷
505
Ðèñ. Ð10.1
u
C
(
t
)
=
1
2
0
C
i t dt
t
( )
ò
+
u
C
(0)
=
–40
e
–2000
t
+ 160
e
–500
t
,
i
3
(
t
)
=
3 + 0,2
e
–2000
t
– 3,2
e
–500
t
À,
i
1
(
t
)
=
3 +
e
–2000
t
– 4
e
–500
t
À.
7.
Ðàñêëàäûâàÿ âûðàæåíèÿ
I
(
p
) íà ïðîñòûå äðîáè, íàõîäèì:
à
)
I p
U
r
Ap B
p
C
p
U
r
p
p
( )
(
)
,
,
(
)
=
+
+
+
+
é
ë
ê
ù
û
ú =
+
+
-
0
2
0
2
1
3
0 25
0 75
1
0 25
3
,
,
p
+
é
ë
ê
ù
û
ú
i t
U
r
e
te
e
t
t
t
( )
[
];
=
+
-
-
-
-
0
3
4
2
á
)
i t
U
r
e
t
( )
(
,
);
,
=
-
-
0
0 5
1 0 25
â
)
I p
U
p
p
p
U Ap B
p
C
p
( )
(
)(
)
=
+
+
+
=
+
+
+
+
é
ë
ê
ù
û
ú =
0
2
2
0
2
2
2
1
2
2
w
w
=
+ +
+
+
-
+
+
é
ë
ê
ù
û
ú
U
p
p
p
0
2
2
2
2
2
3
2 2
4
3
4
2
w
w
w
w
(
)(
) (
)(
)
,
i t
U
t
U
t
U e
( )
cos
(
)
(
)
sin
=
+
+
+
+
-
+
-
3
4
2 2
4
3
4
0
2
2
0
2
0
2
2
w
w
w
w
w
w
w
t
.
8.
Ïîäñòàâëÿÿ â âûðàæåíèå
I p
U p
r pL
( )
( )
=
+
èçîáðàæåíèå
U p
U
p
m
( )
=
+ a
âõîäíîãî íà-
ïðÿæåíèÿ, ïîëó÷àåì òîê
I p
U
L p
p
m
( )
(
)(
)
,
=
+
+
a
d
ãäå
d =
r
L
,
i t
U
r
L
e
e
m
t
t
( )
(
).
=
-
-
-
-
a
a
d
Çàâèñèìîñòü
i t
( ) ïðè
d
a
=
2 èçîáðàæåíà íà ðèñ. Ð9.5. Òîê
i
(
t
)
äîñòèãàåò íàèáîëüøåãî çíà÷åíèÿ, ðàâíîãî
U
r
m
2
, ïðè
t
0
1
0 3
=
-
@
d a
d
a
a
ln
, .c
9.
Çàïèñûâàåì óðàâíåíèÿ
r i
L di
dt
M di
dt
U
r i
M di
dt
L di
dt
1 1
1
1
2
0
2 2
1
2
2
0
+
+
=
+
+
=
,
â îïåðàòîðíîé ôîðìå
(
) ( )
( )
,
( ) (
) ( )
r
pL I p
MpI p
U p
MpI p
r
L p I p
1
1
1
2
0
1
2
2
2
0
+
+
=
+
+
=
506
Îòâåòû íà âîïðîñû, ðåøåíèÿ óïðàæíåíèé è çàäà÷
Ðèñ. Ð9.5
è, ââîäÿ îáîçíà÷åíèÿ
d
d
s
1
1
1
2
2
2
2
2
2
1 2
1
=
=
= -
=
r
L
r
L
k k
M
L L
,
,
,
, ïîñëå ïðîñòûõ ïðå-
îáðàçîâàíèé ïîëó÷àåì:
I p
U
pL
p
p
p
I p
U M
L L
1
0
1
2
2
1
2
1 2
2
0
1 2
( )
[
(
)
]
,
( )
[
=
+
+
+
+
= -
d
s
d
d
d d
s
d
d
d d
p
p
2
1
2
1 2
+
+
+
(
)
]
.
Èñïîëüçóÿ òåîðåìó ðàçëîæåíèÿ, íàõîäèì âåëè÷èíû
i t
U
r
U
bL
p e
p
p t
1
0
1
0
1
2
2
1
1
1
1
1
( )
=
+
+
æ
è
çç
ö
ø
÷÷
-
+
æ
è
çç
ö
ø
÷
s
d
s
d ÷
é
ë
ê
ù
û
ú
= -
-
e
U t
U M
bL L
e
e
p t
p t
p t
2
1
2
2
0
1 2
,
( )
(
),
ãäå
b
k
p
b
=
-
+
= -
+
±
(
)
,
[ (
)
]
.
,
d
d
d d
d
d
s
1
2
2
2
1 2
1 2
1
2
4
2
 ÷àñòíûõ ñëó÷àÿõ òîêè
i t i t
1
2
( ), ( ) âûðàæàþòñÿ áîëåå ïðîñòûìè ôîðìóëàìè.
Íàïðèìåð, ïðè
d
d
d
1
2
=
=
èìååì
i t
U
r
U
r
e
e
i t
U
r
k
t
k
t
1
0
1
0
1
1
1
2
0
2
2
( )
,
( )
=
-
+
æ
è
ç
ç
ö
ø
÷
÷
=
-
+
-
-
d
d
1 2
1
1
r
e
e
k
t
k
t
-
+
-
-
+
æ
è
ç
ç
ö
ø
÷
÷
d
d
.
Ïðè
k
=
1, ò. å. ïðè
M
L L
2
1 2
=
, ïîëó÷àåì
i t
U
r
U
r
e
i t
U
M
e
p t
p t
1
0
1
0
1
2
1
2
2
0
1
2
1
1
( )
,
( )
(
)
.
=
-
+
= -
+
d
d
d
d
d
Åñëè ïðè ýòîì
r
r
r
1
2
=
=
è
L
L
L
1
2
=
=
, òî
i t
U
r
e
i t
U
r
e
t
t
1
0
2
2
0
2
1 1
2
2
( )
,
( )
.
=
-
æ
è
ç
ç
ö
ø
÷
÷
= -
-
-
d
d
10.
Ââåäåì îáîçíà÷åíèÿ
G
(
j
w
)
=
G
(
w
) exp [
j
a
(
w
)]
, N
(
j
w
)
=
N
(
w
) exp [
j
b
(
w
)]. Òî-
ãäà ïîëó÷àåì:
G j
H j
e
G j
H
j
e
G
e
e
j t
j t
j
j t
( )
( )
(
)
(
)
( )
( )
w
w
w
w
w
w
w
a w
w
'
'
+
-
-
=
-
2
j N
e
G
e
e
j N
e
G
j
j
j t
j
w w
w
w w
w
b w
a w
w
b w
( )
( )
( )
( )
( )
( )
( )
+
-
=
=
-
-
-
2
[
]
( )
( )si
[
( ) ( )]
[
( ) ( )]
e
e
j N
G
j t
j t
w a w b w
w a w b w
w w
w
+
-
-
+
+
-
=
2
n[
( )
( )]
( )
( )
( )
( )
( )
w a w b w
w w
w
w w
a w
b w
w
t
N
G
e
N
e
e
j
j
j t
+
-
=
=
é
Im
ë
ê
ù
û
ú =
é
ë
ê
ù
û
ú
Im
G j
N j
e
j t
( )
( )
.
w
w
w
w
11.1. ×àñòîòíûå õàðàêòåðèñòèêè íåïåðèîäè÷åñêèõ ñèãíàëîâ
ÂÎÏÐÎÑÛ
5.
Êàê âèäíî èç àìïëèòóäíîé ÷àñòîòíîé õàðàêòåðèñòèêè
U
(
w
)
=
2
U
0
| (
sin
a
w)/w |
ïðÿìîóãîëüíîãî èìïóëüñà äëèòåëüíîñòüþ 2
a
(ñì. ðèñ. Ð11.1), ïðè óìåíüøåíèè
Îòâåòû íà âîïðîñû, ðåøåíèÿ óïðàæíåíèé è çàäà÷
507